Challenge • Encouraging mathematics learning that goes beyond ‘the basics’ – Learning and teaching ‘the basics’ in mathematics is hard enough – Also need to aim for • Deep and robust understanding – not rules without reasons • Flexible problem solving • Ability to use mathematics • Knowledge of mathematics as a way of thinking

• Focus on mathematical reasoning – in classrooms – in the Australian Curriculum

ACER Research Conference 2010

Australian Curriculum (2010 March)

ACER Research Conference 2010

Australian Curriculum (2010 March) • Proficiency Strands – – – –

Understanding Fluency Problem Solving Reasoning

• Content Strands – Statistics and Probability – Measurement and geometry – Number and Algebra

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Mathematics content and process • 1970s: first explorations of mathematical modelling • 1980s: NCTM agenda for action – promoting problem solving – by 1988, prominent in every state document in Australia

• 1990s: Working Mathematically strand (many variations) – strand includes problem solving, reasoning, communication, technology…..

• 2010: Proficiency strands – reasoning and problem solving

• ALWAYS: Want WM a reality in most classrooms – an elusive goal – TIMSS video study 1999 a reality check – “shallow teaching syndrome” ACER Research Conference 2010

Plan • A little background on the “shallow teaching syndrome” • Several studies looking at this – Survey of mathematics education leaders – Review of mathematics textbook problems – Review of the nature of reasoning and explanation in textbooks

• Video of classroom highlighting some aspects of reasoning • Discussion of reasoning strand in Australian Curriculum

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SHALLOW TEACHING SYNDROME: Textbook Survey

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TIMSS Video Study 1999: Methodology • Video-taped 638 Year 8 lessons from the seven countries – Video-taping spread across school year – Lesson selected at random; little warning given – Random sample of schools and (volunteer) teachers

• Australia: 87 schools, 1950 pupils • Comparative analysis of many characteristics Hiebert, J., Gallimore, R., Garnier, H., Givvin, K.B., Hollingsworth, H., Jacobs, J., Chui, A.M.-Y., Wearne, D., et al (2003). Teaching Mathematics in Seven Countries: Results from the TIMSS 1999 Video Study (NCES 2003-013). Washington DC: National Center for Education Statistics. Hollingsworth, H., Lokan, J. & B. McCrae, B. (2003) Teaching Mathematics in Australia: Results from the TIMSS 1999 Video Study. Melbourne: ACER. ACER Research Conference 2010

Australia in comparison • Many strong features of Australian lessons – especially teachers (overwhelming strongest point from our survey)

• Shallow Teaching Syndrome – in comparison Australian Year 8 lessons exhibited: – A very high percentage of problems that were very close repetitions of previous problems – A very high percentage of problems that were of low procedural complexity (e.g. small number of steps, not bringing different aspects together) – General absence of mathematical reasoning – [Somewhat low on percentage of problems using real life contexts]

• Aspects of absence of reasoning – No lessons contained ‘proof’ (even informal) – Very few problems requiring students to ‘make mathematical connections’ – When problems required connections, these were not emphasised in discussing solutions • Often give the result only • or often focus on the procedures employed ACER Research Conference 2010

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Study of Australian Year 8 textbooks • Chose market-leaders from 4 states and 5 more from one state – gives a picture of work presented to many students

• Chose 3 topics – Adding and subtracting fractions – Solving linear equations – Triangles and quadrilaterals

• Classified as in video study (modified from lesson to book) looking at repetition, complexity and requirement for formal or informal proof • Results: – Broadly consistent with Video Study – Variation of prevalence between textbooks – Tendency for revision of procedures only, and not reasoning

• Work of Jill Vincent and Kaye Stacey – several papers in MERJ ACER Research Conference 2010

Applying procedure learned in one context in another context (“application”) Percentage of problems in sample of eighth-grade textbooks that were applications 80

Percentage of problems

70 60 50 40 30 20 10 0 A

B

C

D

E Textbook

F

G

H

I

Adding/subtracting fractions Solving linear equations Triangles and quadrilaterals ACER Research Conference 2010

Number of triangle and quadrilateral problems in sample of eighth-grade textbooks that were proofs 50 45

Number of problems

40 35 30 25 20 15 10 5 0 A

B

C

D

Worked examples

E Textbook

F

G

H

I

Problems within problem sets

Other problems are mainly calculations and ‘naming’ (stating concepts) (e.g, given two angles, find the third; identify type of triangle). ‘Proof’ here includes informal argument. ACER Research Conference 2010

SURVEY OF LEADERS

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Shallow Teaching Syndrome – is it a real and concerning phenomenon? • Is there (ten years later) – An undesirably high prevalence of repetition? – An undesirably high prevalence of low complexity problems? – An undesirable absence of mathematical reasoning?

• Survey of 20 mathematics education leaders – thank you! – Participants • From departments and systems • Textbook authors • From mathematics associations – Very few differences between these groups • association people tend to express more of teachers’ view – Sometimes difficult to find people with substantial mathematics expertise in departments and systems ACER Research Conference 2010

Examined lessons, textbook extracts, student work, population assessment data Lessons from Several sources

Rule for number of visible faces as function of row length

50o xo 30o

Exercise sets from common textbooks ACER Research Conference 2010

Shallow Teaching Syndrome – is it a real and concerning phenomenon? • Is there (ten years later) – An undesirably high prevalence of repetition? – An undesirably high prevalence of low complexity problems? – An undesirable absence of mathematical reasoning?

Mostly qualified yes Repetition and Complexity: Questions of balance Mathematical Reasoning: Question of ‘what’

In 10 years, computers are the main change

Textbook teaching often blamed ACER Research Conference 2010

Is there too much close repetition at Year 8? • Mixed Response – Yes (55%) – No (11%) – Sometimes (33%)

• Why a lot of close repetition – Balancing act with different balance points for different students • Confidence and fluency VS Boredom • Confidence and fluency VS Lost opportunities for other aspects of mathematics – Other reasons • Following textbook • Behaviour management • To maximise results on common forms of assessment • Unqualified teachers ACER Research Conference 2010

Reasons for perceived high level of repetition in textbooks • Belief that practice makes perfect – a lot of repetition is the best way (63%) • Textbook publishers’ perception of teachers’ preference (31%) • Easier for textbook authors, writing with few resources (26%) • Provides security for (unqualified) teachers (31%) • Provides opportunity for teachers to be selective (31%) • Catering for different needs in mixed ability classes (21%) ACER Research Conference 2010

Procedural complexity

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Identified roles of problems of low complexity • Benefits of using low complexity problems – – – – –

Build confidence because accessible for all students Control over introducing new concepts Catering for all abilities Behaviour management Easy for teachers

• Negative consequences – Lack of connections being made – Limits time for deeper explorations and applications – Disengagement of students because of lack of challenge

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Reasons for perceived high prevalence of low complexity problems • Under-qualified teachers (50%) • Behaviour management (33%) • Use of textbooks (28%) • Easier for teachers and for students (28%) • Entrenched teaching methods (22%) • Low community expectations (22%) • Time issues (17%) • Mixed ability classes (11%) • Perception of a ‘good teacher’(11%)

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Varying opinions on nature of appropriate arguments for Year 8 • Role of (possibly informal) proof following a dynamic geometry investigation of sum of exterior angles of a pentagon – “I would be happy to students just explore these results” – “there are good opportunities for deductive reasoning here” – (proof) “not particularly appropriate for many students” – “I would not do that” – “have students generalise the result to other polygons” – “too obvious to prove”

Sum of exterior angles is 360 degrees ACER Research Conference 2010

Great variety of interpretation of WM terms and identification within lessons – missing shared language or shared goals? • Recognising connections • Applying problem solving strategies • Testing conjectures • Generalising • Developing arguments 50o

• Following arguments xo

• Selecting and translating between representations

30o

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What reasoning is there in textbooks?

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Study of reasoning in 9 Yr 8 textbooks • Overall aim – examine the mathematical reasoning and proofs presented to Australian students • Specific Aim – examine the reasoning that would be evident in “textbook teaching”

• Examine reasoning in explanations provided – look at proofs, verifications, deductions, justifications, explanations etc (Video study ‘PVD’) – formal and informal (well, mainly informal, of course!)

• Limitations – textbook student experience – not looking at reasoning implied in exercises ACER Research Conference 2010

Explanations and proofs • Explanation is not the same as proof – some proofs do ‘explain’ and some do not – some explanations are proofs and some are not

“I remember one theorem that I proved, and yet I really couldn’t see why it was true. It worried me for years and years…. I kept worrying about it, and five or six years later I understood why it had to be true. Then I got an entirely different proof. Using quite different techniques, it was quite clear why it had to be true.” (Michael Atiyah, p. 151 in Mancosu, Jørgensen & Pedersen, 2005) ACER Research Conference 2010

Sierpinska (1994)

Proofs

Explanations

Scientific explanation

Inference (Science)

Deduction (Mathematics)

Didactic explanation

Example

Model

Visual representation

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Method • Chose market-leaders from 4 states and 5 others from one state – gives a picture of explanations presented to many students

• Chose 7 topics where we expected mathematical reasoning was required – state differences, so not every topic was in every textbook

• For each topic, we examined all the explanations, justifications and reasoning presented explicitly – did not examine implicit reasoning in worked examples, or when solving exercises – 53 treatments of 7 topics with 69 distinct explanations

• Each explanation was examined very carefully to identify the nature of the reasoning that supported the critical steps of the argument • From these examples, a list of the modes of reasoning was created – compared to other schemes (e.g. Harel and Sowder; Sierpinksa; Blum & Kirsch) – Stacey & Vincent (2009) Educational Studies in Mathematics ACER Research Conference 2010

Results: Identified 7 modes of reasoning • deduction using a general case • deduction using a specific case • deduction using a model • concordance of a rule with a

Rule for number of visible faces as function of row length – may be Deduction using general case model Deduction using specific case Experimental demonstration

• experimental demonstration • qualitative analogy • appeal to authority

Some explanations are easy to categorise; some depend on interpretation of writer’s intention ACER Research Conference 2010

Notes agreement with rule; does not deduce the rule (but it could)

Division of fractions

Model 1: Quotition

Model 2: Partition

Text book X: Concordance of a rule with model

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A working spreadsheet

Textbook Y: Experimental demonstration OR appeal to authority

Textbook Y: Deduction using a specific case ACER Research Conference 2010

Textbook Y: Qualitative Analogy (reason 3) (but analogical reasoning applies much more widely e.g. using a model) Not a model of multiplication – only signs being “multiplied” e.g. +x-=(It could be made into a model using velocity of movie star and velocity of film to give apparent velocity of star on film ACER Research Conference 2010

Textbook Z: Deduction using a general case (guided discovery)

Textbook Z: Experimental demonstration

Textbook Z: Deduction using a general case (guided discovery)

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http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/mcd/m47505p.htm ACER Research Conference 2010

Volume of a sphere (Mathematics Developmental Continuum)

http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/mcd/M47505sp.htm

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Observations on explanations • Very little “rules without reasons” (but certainly some) – not what might be expected of “textbook teaching”

• Nearly all (deductive) explanations were ‘correct’ but incomplete, omitting Not necessarily bad!

– basic reasoning and linking commentary – difficult cases and subtle points.

• Explanations brief and unlikely to stand alone – students must rely on teachers to elaborate – teachers need to know how to do this elaboration (MPCK)

• Purpose appeared to be to derive the rule in preparation for practice exercises – little use of explanatory model etc as a thinking tool – revision chapters tend to focus on procedures, not reasons ACER Research Conference 2010

After finding the rule, the reasoning is usually not revisited ACER Research Conference 2010

Conclusions about reasoning • Textbooks used a variety of modes of reasoning • Variety is between topics, as well as between textbooks • Four modes of reasoning are unacceptable from a mathematical point of view (to varying degrees): – – – –

qualitative analogy appeal to authority experimental demonstration concordance of a rule with a model

• Mathematically unacceptable reasoning may or may not be useful pedagogically – Do students understand the differences – mathematical reasoning they should learn OR local pedagogical purpose (e.g., to help remember). – Are students presented with acceptable (appropriate) reasoning? ACER Research Conference 2010

Lesson study video clips Making reasoning more than a ritual before the rule

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APEC Lesson study • Asia-Pacific Economic Cooperation (APEC) EDNET project on Classroom Innovation through Lesson Study • Organisers: Drs Masami Isoda, Shizumi Shimizu, Maitree Inprasitha, Suladda Loipha, Alan Ginsburg • Website with many lessons to download from countries around Pacific Rim

http://hrd.apec.org/index.php/Area_of_the_Circle_Grade_5_(Japan)

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Yasuhiro Hosomizu 5th grade APEC Lesson study video Dec 2006 • Area of circle – sector rearrangement method had been discussed in previous lesson • Focus of lesson – to find formula for area of circle by rearranging segments of circles into (approximate) shapes with known areas – May use triangles, parallelograms, rectangles, trapezoids because these are the shapes with known area formulas

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Extract from lesson plan (from APEC wiki)

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36.45 – 39.45 mins Area = base x height = diameter x circumference ÷ 4 = 20 x circumference ÷ 4 = 5 x circumference Alternatives: Diameter ÷ 4 x circumference Radius ÷ 2 x circumference Radius x radius x pi

Yasuhiro Hosomizu 5th grade

• Formulas from parallelogram – – – – – –

Half of circumference x radius Radius x pi x radius Diameter x pi ÷ 2 x radius pi x diameter ÷ 2 x radius pi x (radius x 2) ÷ 2 x radius pi x radius x radius (video at about 20 mins)

Also alternate between pi and 3.14

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Observations • Some very well prepared students, with creative ideas • Value in repeating the reasoning and argument with different shapes – even these excellent students need to revisit the ideas • Even expert teacher finds it hard to use children’s explanations productively – Shift by one boy between the general and particular (at 36 mins) makes it very hard for other students to follow reasoning

• Often even harder to use wrong reasoning productively • Focus on deep mathematical principles in lesson – e.g. using the area formulas that you know • Secondary aim of expressions: Difficulty and multiple steps in moving between expressions

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Reasoning in the Australian Curriculum

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Australian Curriculum (2010 March)

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Australian Curriculum Proficiency Strands • Understanding • Fluency • Problem Solving – Students develop the ability to make choices, interpret, formulate, model, and investigate problem situations, and communicate solutions effectively

• Reasoning – Students develop increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising

• Also “general capabilities” across subjects which include “thinking skills” and “creativity” (p. 5) ACER Research Conference 2010

Mathematics content and process – concerns • In the survey of leaders, all were concerned (although details differ) and all recommend – Better resources – Teacher education – Therefore important that Australian Curriculum provides sound advice

• Different nature of four proficiency strands means they need different treatment – Understanding and fluency – inherent part of learning content well – Problem solving and reasoning – more than this • Important outcomes of learning, independent of content • Part of the fabric of any real mathematics lesson • Also contributing to learning content

• Dilemma of separation from content VS integration with content – in class and in curriculum specification – Need to identify relevant goals for broad age groups Everywhere and nowhere ACER Research Conference 2010

Reasoning and PS are not just “forms” of classroom interaction (e.g. discussion) • Reasoning is not evident in the form of classroom interaction, but in the substance • External evidence of reasoning – Classroom discussion • Find reasons and arguments • Compare reasons and arguments • Analyse reasons and arguments – Writing arguments • Consolidate reasoning • Check reasoning

• Reasoning happens when students work by themselves too! ACER Research Conference 2010

To encourage reasoning: • Some indication of how reasoning develops, and characteristics of different stages – That both empirical and deductive argument are present from the early years of school – Some indication of developing complexity (e.g. length of argument) – Algebra provides a language for generality, but this does not mean that students do not make general arguments before this (“deduction using a specific case”)

• Consistent examples of appropriate mathematical reasoning at each level – Example: the role of definitions in mathematics – Give examples which reinforce reasoning, not just practice rules ACER Research Conference 2010

Argument from definition • A very strong feature in mathematical reasoning • Must be strengthened in curriculum document by attention to detail

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Australian Curriculum - Circles • Year 1 (p. 10)

describe shapes

– “saying circles are round”

• Year 2 (p.14) – “sorting circles, triangles and rectangles and saying the grouping is based on the number of straight sides” – Year 5 (p.33) “noting similarities such as all quadrilaterals have 4 straight sides”

• Year 7 ( p.46) – [using pairs of compasses for construction]

• Year 8 (p 51 et seq) – Considerable circle geometry ACER Research Conference 2010

Australian Curriculum Yr 1 (p.10) “saying that circles are round”

http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/space/SP25001P.htm ACER Research Conference 2010

even + even = even Definition of even (and hence odd) numbers - Multiple of 2 - Last digit is 2, 4, 6, 8, 0 - Is made up from pairs

Empirical (experimental demonstration) – look at examples 1+1=2

2+4=6 3+3=6

2 + 6 = 8 13 + 5 = 18 12 + 4 = 16 14 + 5 = 19 ACER Research Conference 2010

Argument available to young students – but it depends on definition

Note “seeing the general in the particular” (Deduction using a special case) ACER Research Conference 2010

http://www.fgworld.co.uk/Schooldays/Mary%20Hines/st% 20mikes%20walking%20day%20kids.jpg ACER Research Conference 2010

Explanation? To prove that the sum of two even numbers is even Let a and b be even numbers Then a 2m for m ∈ b 2n for n ∈ Therefore a + b = 2m + 2n = 2(m + n) Since (m + n) ∈ , a + b is an even number Definition of even (and hence odd) numbers - Multiple of 2 - Last digit is 2, 4, 6, 8, 0 - Is made up from pairs

T o s h o w t h e s u m

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Summary • Understanding and fluency strands – These relate to the quality of learning content goals

• Problem solving and reasoning strands – These relate to the quality of learning content goals – Also more elusive but fundamental goals

• Repetition and low complexity : Out of balance! – “justified” by the fluency goal – but there are many other reasons

• Reasoning – Many types of reasoning in good math’l didactic explanation – Highlight the mathematical ones, use others knowingly – Review the reasoning – don’t make it just a starting ritual

• Draft Australian curriculum – aim to make maths teaching better – Clarify goals for reasoning throughout – Build towards reasoning in every aspect ACER Research Conference 2010

Thank you The reported studies are joint work with Dr Jill Vincent

ACER Research Conference 15 – 17 August, 2010 [email protected]